Weird Limits

Relevant wiki: L'Hôpital's Rule

$\begin{aligned} L & = \lim_{n \to \infty} \left(2 \sqrt[n]{1000}-1\right)^n \\ & = \lim_{n \to \infty} \left(2\times 10^\frac 3n -1\right)^n \\ & = \lim_{n \to \infty} \exp \left(\ln \left(2\times 10^\frac 3n -1\right)^n \right) & \small \color{#3D99F6} \text{where }\exp(x) = e^x \\ & = \exp \left( \lim_{n \to \infty} n \ln \left(2\times 10^\frac 3n -1\right) \right) & \small \color{#3D99F6} \text{Let }x = \frac 1n \\ & = \exp \left( \lim_{x \to 0} \frac {\ln \left(2\times 10^{3x} -1\right)}x \right) & \small \color{#3D99F6} \text{A 0/0 case, L'Hôpital's rule applies.} \\ & = \exp \left( \lim_{x \to 0} \frac {2(3\ln 10) 10^{3x}}{2\times 10^{3x} -1} \right) & \small \color{#3D99F6} \text{Differentiate up and down w.r.t. }x \\ & = \exp \left( \ln 10^6 \right) \\ & = \boxed{1000000} \end{aligned}$

Note:

$\begin{aligned} f'(x) & = \frac {d}{dx} \ln ({\color{#3D99F6}2\times 10^{3x}-1}) & \small \color{#3D99F6} \text{Let } u = 2 \times 10^{3x} - 1 \\ & = \frac {d}{dx} \ln \color{#3D99F6} u \\ & = \frac 1u \times \frac {du}{dx} \\ & = \frac 1{2\times 10^{3x}-1} \times \frac d{dx} (2 \times {\color{#3D99F6}10}^{3x} - 1) & \small \color{#3D99F6} \text{Note that } 10 = e^{\ln 10} \\ & = \frac 1{2\times 10^{3x}-1} \times \frac d{dx} (2 {\color{#3D99F6}e}^{3x\color{#3D99F6}\ln 10} - 1) \\ & = \frac {2(3\ln 10)e^{3x\ln 10}}{2\times 10^{3x}-1} \\ & = \frac {(6\ln 10)10^{3x}}{2\times 10^{3x}-1} \end{aligned}$